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# Scalar and Vector Operations

### displacement vector and distance

Hello I need help answering these problems. Peter noticed a bug crawling along a meter stick and decided to record the bug's position in five-second intervals. After the bug crawled off the meter stick, peter created the table shown. 1. What is the displacement of the bug between t = 0.00 s and t = 20.0 s? 2. what is the to

### Divergence Theorem for a Function

1. Check the divergence theorem for the function: V(vector)= r^2*cos(theta)(r(hat))+r^2*cos(phi(Theta(hat)) -r^2*cos(theta)*sin (phi)(phi(hat)) Use as the volume one octant of the sphere of radius R.

### Divergence of a vector function

Given vector function V = xyi +2yzj+3zxk, what is the divergence of this function? is the magnitude of the divergence greater at the origin or at (1,1,1)

### Relative Motion - A boat travels up a river a distance d and then comes back to the origin

A boat travels up a river a distance d and then comes back to the origin. The maximum speed of the boat relative to the water is v. And the river has scalar speed u (u < v). What is the minimum time for this trip?

Calculate the gradient Vf of the following functions, f(x,y,z) a. f = x^2 + z^3 b. f = ky where k is a constant c. f = r = (x^2 + y^2 + z^2)^1/2 Hint use the chain rule d. f = 1/r See attachment for better symbol representation

### Curves in euclidean 3-space

In Euclidean three-space, let p be the point with coordinates (x,y,z) = (1,0,-1). Consider the following curves that pass through p: Curve 1: xi (λ) = (λ, (λ-1)2, - λ) Curve 2: xi (μ) = (cos μ , sin μ, μ-1) Curve 3: xi (σ) = (σ2, σ3 + σ2, σ) The curves are parametrized by the parameters that v

### Plane Wave Propagation

Please see the attached file for a full description of the problem. Consider a unit vector s = , and a plane perpendicular to s at a distance d from the origin. a. For all points r within this plane, show that r.s = x sz + y sy + z sz = d. b. Show that the scalar function f (x, y, z, t) = exp(i 2 pi (xsx + ysy + z sz)/L -

See attached file for full problem description. Derive the gradient formula for a potential in both cylindrical and spherical coordinates. Also, derive the divergence formula of the vector field in both coordinates.

### Finding Vector Field

See attached file for full problem on vector fields and scalar/cross products. Solve the attached question

### Phasor Calculation

Please see the attached file for calculation of a phasor vector.

### Finding the center of mass of a system of masses

Masses are placed on the x-y plane as shown. Find the center of mass of this system. See attached file for full problem description.

### Applications of the Stokes Theorem

Stokes Theorem. See attached file for full problem description. 1. compute the line integral where F = (yz^2 - y)i + (xz^2 + x)j + 2xyzk where C is the circle of radius 3 in the xy-plane, centered at the origin, oriented counterclockwise as viewed from the positive z -axis. 2. Given F =yi - xj + yzk and the region S determ

### Addition of spin angular momenta

Addition of spin angular momenta Consider the addition of two spin -1/2 angular momenta, S(1) and S(2) 1. How many states are there in the product basis? 2. If J = S(1) + S(2), what are the possible eigenvalues of the dot product of J and J? 3. By using the recursive algorithm construct all the Clebsch-Gordan coefficient

### Moment of inertia of a system of masses.

An object is made up of three masses connected by massless rods of fixed length. Mass A is located at (30.0 cm, 0 cm) and has a mass of 250 grams, mass B is located at (0 cm, 30.0 cm) and has a mass of 350 grams, mass C is located at (-30.0 cm, 0 cm) and has a mass of 450 grams. What is the moment of inertia of this object about

### Eigenvalues, eigenvectors, and time evolution.

Dear Mitra, I in fact wrote the wrong matrix but I am still confused after PART A. (PART A) The actual matrix is: H = 1 2 0 2 0 2 0 2 -1 Where the eigenvalues are E1 =0, E2=3hw, E3=-3hw and as you said the trace(H) =0 = sum of eigenvalues. I also found the eigenvectors using Hx = Ex and they w

### Westward component

A soccer player carries the ball for a distance of 40.0 m in the direction 42.0 west of south. Find the westward component of the ball's displacement. 26.8 m 29.7 m 36.0 m None of the other choices is correct.

### orthogonality condition is preserved under orthogonal transform

Part i) Show that if u & v are orthogonal, then the transformed vectors U = Au & V = Av under the linear (orthogonal) transformation (characterised by the orthogonal matrix A) are themselves orthogonal. I think this can be done using pythagoras theorem but am not sure how to begin, please help! Part ii) shSw that the orthogon

### Problems involving scalar and vector operations

The eastward component of vector A is equal to the westward component of vector B and their northward components are equal. Which one of the following statements is correct for these two vectors? Choices: Vector A is parallel to vector B Vector A is anti-parallel to vector B The magnitude of vector A is equal to the mag

### Direction relative to ground

A plane has an airspeed of 142 m/s. A 30 m/s wind is blowing southward at the same time as the plane is flying. What must be the direction of the plane in order to move due east relative to the ground? Answer choices: 78.1 degrees north of east 11.9 degrees north of east 77.8 degrees north of east 12.2 degrees north of

Vector A=6 m and points 30 degrees south of east. Vector B=4 m and points 30 degrees south of west. Find the resultant vector A+B. Answer Choices: 3.3 m at an angle of 71 degrees south of east. 5.3 m at an angle of 71 degrees south of east 3.3 m at an angle of 19 degrees south of east 5.3 m at an angle of 19 degrees sou

### The answer to Resultant Force

Three forces are acting on the origin. F1 has a magnitude of 70 N and is 30 degrees north of east. F2 has a magnitude of 70 N and is 30 degrees north of west. F3 has a magnitude of 70 N and is along the south direction. Find the resultant force acting on the origin.

### Vector Transformation for Spatial Coordinate Systems

Consider a two-dimensional spatial coordinate system S' whose coordinates (u,v) are defined by x = u + v y = u - v in terms of the coordinates of a Cartesian coordinate system S. Suppose you are given a vector in S whose contravariant components are Am = (2,8). Determine the contravariant components of this ve

### Velocity of a charge moving in a magnetic field

The problem states: A particle with a charge of -5.90 * 10^-9 C is moving in a uniform magnetic field B(vector) = -(1.25 T) k (hat). The magnetic force on the particle is measured to be F(vector) = -(4.00 * 10^-7 N) i(hat) + (7.60 * 10^-7 N) j(hat). 1. Are there components of the velocity that are not determined by the

### Vectors and Unit Vector

The vector A has a magnitude of 3.00 meters. It makes an angle of 30° with respect to the positive x-axis. The vector B has a magnitude of 3.00 meters, but points along the y-axis. What are the horizontal and vertical components of ? a) Ax = 2.60, Ay = 1.50 b) Ax = 1.50, Ay = 2.60 c) Ax = 3.00, Ay = 3.00 d) Ax = -2.

### Vector problem

So here is the question Let A= {10, 30°} be a vector of magnitude 10 ( in some units) and pointing in a direction of 30°counter-clockwise from positive x. Let B = { 7, 225°degree} Find x and Y componens of these two vectors A { AX= AY= B { Bx= BY= Find x and y components of the sum S= A+B S= { Sx= S

### Resultant Vectors of 3 Displacement Vectors

How do I find the resultant of the three displacement vectors in the drawing by means of the component method? If the magnitudes of the vectors are A= 5 m B=5 m and C= 4 m?

### Magnitude of Displacement Vector

An ocean liner leaves New York and travels 18.0 degrees north of east for 155 km. How far east and how far north has it gone? How do I figure out the magnitudes of the ships displacement vector in the directions due east and due north?

### Magnitude and Direction of a Resultant Vector

If Displacement vector A points due east and has a magnitude of 2.00 km. Displacement vector B points due north and has a magnitude of 3.75 km. Displacement vector C points due west and has a magnitude of 2.50 km. Displacement vector D points due south and has a magnitude of 3.00 km. How can I find the magnitude and direction (

### Pair of forces

Consider a pair of forces, one having a magnitude of 20 N, and the other having a magnitude of 12 N. What maximum net force is possible for these two forces? What is the minimum net force possible?

### electric field and potential of point like charges.

(See attached file for full problem description) --- Two equal, positive charges, q = 2.0 micro Coulomb ; are located on the x-axis- one at +0.3m and the other at -0.3m. A third charge Q = +4.0 micro Coulomb is located on the y-axis at +0.4m 1) Find the magnitude & direction of the resultant (net) force on Q. 2) Find